Question

Which among the following functions is not injective

• A. $f\left(x\right)=|x+1|,x\in [-1,\infty )$
• B. $g\left(x\right)=x+\frac{1}{x},x\in (0,\infty )$
• C. $h\left(x\right)=x^2+4x-5,x\in (0,\infty )$
• D. $k\left(x\right)=e^{-x},x\in [0,\infty )$

Answer 1

The correct option is B $g\left(x\right)=x+\frac{1}{x},x\in (0,\infty )$

$f\left(x\right)=|x+1|,x\in [-1,\infty )$



$h\left(x\right)=x^2+4x-5,x\in (0,\infty )$

$k\left(x\right)=e^{-x},x\in [0,\infty )$

Clearly, from above graphs $f\left(x\right),h\left(x\right),k\left(x\right)$ using horizontal line test in the given intervals of each function, we can say that
they are injective functions in the given interval.
For $g\left(x\right)=x+\frac{1}{x},x\in (0,\infty )$
$g^{\prime }\left(x\right)=1-\frac{1}{x^2}=\frac{x^2-1}{x^2}$
Clearly, $g^{\prime }\left(x\right)$ is not continuously positive or continuously negative. Hence $g\left(x\right)$ is increasing for some interval $\left(\right(-\infty ,1)\cup (1,\infty \left)\right)$and decreasing for some interval $\left(\right[-1,0)\cup (0,1\left]\right)$ . So it is a many-one function.